Exact solution of three-point functions in critical loop models

TL;DR

This paper presents an exact formula for three-point functions in critical loop models, validated through multiple methods, bridging conformal field theory, lattice models, and probability theory.

Contribution

It introduces a novel exact formula for three-point functions in critical loop models and demonstrates its validity using three distinct approaches.

Findings

The formula accurately predicts three-point functions in critical loop models.

Validation through conformal bootstrap, transfer-matrix, and probabilistic methods.

Establishes a deep connection between different theoretical frameworks in 2D statistical physics.

Abstract

We propose an exact formula for three-point functions on the sphere in critical loop models with primary fields $V_{(r,s)}$ characterized by $2r$ legs and a parameter \(s\) that describes diagonal fields for $r=0$ and the momentum of legs for $r>0$. We demonstrate its validity in three ways: the conformal bootstrap method for 4-point functions, a transfer-matrix study of the lattice model, and a probabilistic method based on conformal loop ensemble and Liouville quantum gravity. This work provides a crucial missing piece for solving critical loop models and reveals a deep unity between three fundamental approaches to 2D statistical physics: transfer matrix, conformal field theory, and probability theory.